Fractional Powers
Fractional powers refer to the concept of raising a number to a fraction as an exponent. For example, taking the square root of a number is equivalent to raising it to the power of 1/2. Fractional powers can also be used to represent roots, such as the cube root or fourth root of a number.
3 Key excerpts on "Fractional Powers"
eBook - ePubBasic Math Concepts
For Water and Wastewater Plant Operators
- Joanne K. Price(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...positive: 2 − 2 3 4 = 3 4 2 2 5 2 3 − 3 2 − 2 = 5 2 3 3 2 2 3 2 5 2 2 − 3 = 3 2 5 2 2 3 Example 2: (Negative Exponents) □ Write the following terms in expanded form using factors only (no. exponents): 10 x − 2 ; 2 5 • 4 − 2 ; 6 0 x 3 x − 2 10 x –2 = 10 x 2 = 10 (x) (x) 2 5 4 − 2 = 2 5 4 2 = (2) (2) (2) (2) (2) (4) (4) 6 0 x 3 x − 2 = 6 0 x 3 x 2 =[--=PLGO-SEPARATOR=--. ](1) (x) (x) (x) (x) (x) Example 3: (Negative Exponents) □ Complete the following calculations: (0.785)(60 2)(20); and. (3 2)(2 -3)(6 2) (0.785) (60 2) (20) = (0.785) (60) (60) (20) = 56, 520 3 2 2 − 3 6 2 = (3) (3) (6 3) (6 3) (2) (2) (2) = 40.5 ZERO EXPONENTS When a number has a zero exponent, it is always equal to one: 5 0 = 1 or x 0 = 1 DIVIDE. OUT FACTORS WHENEVER POSSIBLE As with other calculations, divide out common factors whenever possible. This makes the calculation less cumbersome. 13.3 FRACTIONAL EXPONENTS SUMMARY 1. A root is a number which, when multiplied together two (or more) times, equals the original number. A square root is a number which, when multiplied together twice, equals the original number. For example, the square root of 64 is 8, since 8 × 8 = 64. A cube root is a number which, when multiplied together three times, equals the original number. For example, the cube root of 8 is 2, since 2 × 2 × 2 = 8. 2. A fractional exponent indicates a root is to be taken. The denominator of a fracitonal exponent determines which root is to be taken. 3. A fractional exponent of 1/2 indicates that a square root is to be taken. A square root may also be written as a radical: x 1 / 2 = x 4. A fractional exponent of 1/3 indicates that a cube root is to be taken. A cube root may also be written as a radical: x 1 / 3 = x 3 5. The numerator of the fractional exponent is the power of the base. For example: This may be written using a radical as: 8 2 3 THE ROOT OF A NUMBER A root is a number which, when multiplied together a given number of times, equals the original number...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Rule 2:, so when dividing, subtract the exponents. 3. Rule 3: (2 3) 2 = (8) 2 = 64 = 2 6, and 3 × 2 = 6, so when raising a power to a power, multiply the exponents. Roots Roots can be thought of as the opposite of powers. If 3 4 = 81, we say, “3 to the fourth power is 81.” The root sentence for this would be “The fourth root of 81 is 3,” and in symbols it is written as. So this “root” is actually answering the question, “What number repeated as a factor four times gives us 81?” The symbol for root is, called a radical, where the number for which we are finding a root (X in the radical below), called the radicand, goes under the radical sign and the root we are finding (n in the radical below), called the index, is inserted as a smaller (in size) number at the left of the radical sign. Because the most common index is 2, we don’t even bother to write the little 2 on the radical. We just know that has index 2, and it is called a square root. If the index is 3, we do write the little 3, so we have, which has the special name of cube root. All of the other roots are called by their indexes: “fourth root,” “fifth root,” and so forth. Fractional Exponents Fractional exponents imply roots. The numerator still indicates power, but the denominator of the fractional exponent indicates root. For example, means either (the cube root of 8 2) or, the square of the cube root of 8. It makes no difference whether you do the root or the power first. In this case, since the cube root of 8 is 2, it is easier to do the root first, and the answer is. Doing it the other way, we would have to remember, which, of course, also gives us 4. Roots of Negative Numbers You can find square roots of positive numbers only. The square root of a negative number is called imaginary because there are no two identical numbers that multiplied together will produce a negative number (two positives multiplied together or two negatives multiplied together will always produce a positive result)...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...By analogy with defining the reciprocal, we need to define what it means to raise a number to the 1 / n th power. To do that, we first need to define ‘taking the n th root’ of a number. Finding the n th root of a number In much the same way as subtraction is the reverse of addition, and division the reverse of multiplication, taking a root is the reverse of raising to a power. For example, because we know that five to the power four is 625, we can say that a fourth root of 625 is five. We can write this using a special symbol as 625 4 = 5. Likewise we can write down that a square root of 25 is five: 25 2 = 5. Because we frequently need to write down the square root of a number, we usually omit the little ‘2’ before the square root symbol: for example 25 for the square root of 25. Note that there can be more than one square root or fourth root of a number. For instance if we square −5 we also get 25 and if we raise −5 to the fourth power we get 625, thus −5 is a square root of 25 and a fourth root of 625. We often write this by saying that the square roots of 25 are ± 25 = ± 5 and the real fourth roots of 625 are ±5. (If we allow complex numbers, there are two more fourth roots of 625. These are ±5 i where i = − 1 but we promised in the Introduction to ignore these.) Expressed generally if a n = b, then a is an n th root of b. Whenever we need to specify only one n th root we require that the root be positive and call it the principal n th root, b n. Provided the original number, b, is positive there is always a principal n th root. Sometimes the principal root is reasonably obvious. For instance, 2 and −2 are square roots of 4 because either of these when squared gives 4. Similarly 3.1 is a square root of 9.61 (another is −3.1) because 3.1 2 = 9.61 and 3.1 is a cube root of 29.791 because 3.1 3 = 29.791. In each case the principal root is the positive value. However, sometimes the root is not obvious at all...
